A189759 Numbers pqr such that pq + pr + qr is prime, where p, q, and r are primes.

30, 42, 66, 70, 78, 105, 114, 130, 154, 165, 174, 182, 222, 231, 238, 246, 255, 273, 282, 285, 286, 310, 318, 345, 357, 366, 370, 385, 399, 418, 430, 434, 442, 455, 465, 474, 483, 494, 498, 518, 555, 561, 574, 582, 595, 602, 609, 618, 642, 645, 651, 663, 665

Offset

1,1

Comments

The number pq+pr+qr is prime only if p, q, and r are distinct. The primes of form pq+pr+qr are in A087054. A prime may have multiple representations as pq+pr+qr; for example, 2*3*13 and 3*5*7 both produce the prime 71.

As mentioned by Ufnarovski and Ahlander, if pq+pr+qr is prime, then the arithmetic derivative (A003415) of pqr is that prime. They conjecture that this sequence and A087054 are infinite.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Mathematica

pqr[nn_] := Module[{p=Prime[Range[PrimePi[nn/6]+1]], i, j, k, n, prod}, Sort[Reap[i=0; While[i++; p[[i]]p[[i+1]]p[[i+2]] <= nn, j=i; While[j++; p[[i]]p[[j]]p[[j+1]] <= nn, k=j; While[k++; prod=p[[i]]p[[j]]p[[k]]; prod <= nn, n=p[[i]]p[[j]]+p[[i]]p[[k]]+p[[j]]p[[k]]; If[PrimeQ[n], Sow[prod]]]]]][[2, 1]]]]; pqr[1000]
Take[Union[Times@@@Select[Subsets[Prime[Range[30]], {3}], PrimeQ[ Total[ Times@@@Subsets[#, {2}]]]&]], 60](* Harvey P. Dale, Dec 29 2011 *)

Crossrefs

Cf. A003415, A087054, A356475.

Sequence in context: A302574 A087248 A249242 * A046401 A302570 A300156

Adjacent sequences: A189756 A189757 A189758 * A189760 A189761 A189762

Keyword

nonn

Author

T. D. Noe, Apr 27 2011

Status

approved